Chains of Modular Elements and Lattice Connectivity

We show that the order complex of any finite lattice with a chain 0̂ < m1 < · · · < mr < 1̂ of modular elements is at least (r − 2)-connected. In [St], Stanley shows that a finite lattice L with a maximal chain consisting entirely of modular elements is supersolvable and therefore graded and EL-shellable in the sense of [Bj1]. Hence, the order complex ∆(L \ {0̂, 1̂}) has the homotopy type of a wedge of top dimensional spheres. In particular, if L has a maximal chain 0̂ < m1 < · · · < mr < 1̂ with each mi modular then its order complex ∆(L \ {0̂, 1̂}) is at least (r − 2)-connected. We generalize this connectivity lower bound to chains of modular elements that are not necessarily maximal. Knowledge about the topological structure of the order complex of a given lattice can be of use, for example, in the study of subspace arrangements and in the study of free resolutions. See for instance [GM], [ZZ], [Bj2], and [GPW]. If instead of determining the homology entirely, one merely is able to prove a connectivity lower bound, this already may provide useful information. For example, connectivity lower bounds for LCM lattices directly translate to upper bounds on the regularity of a monomial ideal; a connectivity lower bound for monoid posets gives a bound on the rate for resolving the residue field over an associated toric ring, i.e. the coordinate ring of an associated toric variety. Terminology as well as specific results in these directions may be found, for instance, in [MS], [HRW], [HW], [Pe] and [PRS]. We assume that the reader is familiar with the basic notions from topological combinatorics. All relevant definitions can be found in [Bj3]. All lattices we consider here will be finite with minimum element 0̂ and maximum element 1̂. One of many equivalent definitions of modularity in a lattice L is that m ∈ L is modular if for each x ∈ L The first author was supported during part of this work by a postdoctoral fellowship from the Mathematical Sciences Research Institute. The second author was supported by NSF grant DMS-0300483.